Fusing Interval Preferences

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1 Fusing Interval Preferences Taner Bilgiç Department of Industrial Engineering Boğaziçi University Bebek, İstanbul, Turkey Appeared in Proceedings of EUROFUSE Workshop on Preference Modelling and Applications, Granada Spain, April 2001, pp Abstract We consider multiple agent settings where each agent reports a partial preference order. We derive a condition, called the TSinterval order property, on the underlying fuzzy weak preference relation which, when satisfied, yields an interval order on the set of alternatives. We then propose a greedy algorithm to aggregate those interval orders. Keywords: Preference Modelling, Interval orders, Aggregation. 1 Introduction In this paper, we are concerned with fusing the preferences of multiple agents with respect to a set of alternatives. The problem might arise in different contexts but our motivation is multiple software agents reporting their own preference orderings in the spirit of Distributed Artificial Intelligence. What is being reported might be preferences over a set of alternatives as well as a partial solution to a common problem (e.g., as in distributed scheduling). What needs to be done is to construct a common linear ordering that is acceptable to all agents. A similar problem has been studied in the context of social choice theory [1] where each agent reports a linear order and this leads to a notorious impossibility theorem. Aggregating fuzzy measures has also received some attention [7], but we conduct an ordinal analysis in the spirit of [2, 5] where fuzzy preferences are purely ordinal. Recently Yager [13] took up a similar problem and came up with a greedy aggregation algorithm for importance ordered linear orderings obtained from multiple agents. In his setting, the agents are reporting linear orderings. We assume, agents can report interval orders. In this paper, we assume that agents are reporting only partial (preference) orders due to possibly limited knowledge and/or computational resources. We assume that the partial order reported is an interval order which is a subset of all partial orders. Asymmetric parts of interval orders satisfy transitivity but the corresponding equivalence (indifference) relation need not be transitive (see e.g. [9, 6, 11]). There may be various reasons why agents are reporting interval orders rather than more general partial orders. One such case is when strict preference as constructed from a fuzzy weak preference relation is interval-valued [3]. Another case is when each agent is trying to solve (part of) a scheduling problem. There are two issues in coming up with aggregation of interval orders. The first is to find an extension of each individual interval order such that information encoded in the interval order is preserved as much as possible. The second issue is to come up with an aggregation of either many interval orders or the linear extensions of many interval orders. In Section 2, we discuss how interval strict preference orders induce a a crisp interval order on the set of alternatives. Particularly, we derive a condition, called TS-interval order property, on the underlying fuzzy preference relation. The property is valid for any continuous De Morgan triple ht;s;ni. In Section 3, we discuss extending interval orders to linear orders with the corresponding jump number problem. In Section 4, we propose a greedy algorithm to ag-

2 gregate interval orders and derive some of the desired properties of the proposed algorithm. Section 5 concludes the paper and formulates some further research questions. 2 Interval Preference Orders Let R be a (crisp) binary relation on a finite set A (i.e., R A A). We usually take R to represent weak preference and therefore R is naturally reflexive and transitive. The strict preference, P, is defined as: ap b () arb and not bra: (1) When the underlying weak preference relation, R is a fuzzy relation (i.e., R : A A! [0; 1]), the strict preference P has to be defined using the language of fuzzy sets [10]. We assume a De Morgan triple, ht; S; Ni represents fuzzy conjunction, disjunction and (strong) negation, respectively where T is a t-norm, N is a strong negation function and S is a t-conorm produced from T via N. We assume all functions are continuous on the unit interval. In [3], in order to define an interval-valued strict preference relation P, the righthand side of the definition given in (1) is interpreted as non-implication. Using the notation x = R(a; b) and y = R(b; a), is written as: (for all x; y 2 [0; 1]): P (a; b) = [P D (a; b);p C (a; b)] = [T (x; N (y));t(s(x; y); S(x; N (y));s(n (x);n(y)))] where, P D, is the DNF representation of nonimplication, and P C is the corresponding CNF part. Using the Boolean Disjunctive and Conjunctive Normal Forms and their separation in fuzzy set theory for any De Morgan triple comprised of a continuous triangular norm, co-norm and a strong negation function, one can construct interval-valued preference structures. Hence any higher order definition based on the fuzzy weak preference relation R is intervalvalued [2, 12, 3]. The motivation is as follows, whenever the underlying preference relation is fuzzy and higher order constructs from it (like strict preference, indifference, incomparability) fall prey to impossibility theorems. To avoid that framework, we propose an interval-valued framework where the well known transitivity and connectivity properties on the underlying weak fuzzy preference relation lead to a linear strict order. Then a crisp binary relation on A is defined as follows: Definition 1 A (crisp) ordering relation χ is defined as (for all a; b 2 A) a χ b if and only if P D (a; b) >P C (b; a): (2) The relation χ means a is strictly preferred to b if and only if the interval P (a; b) is strictly to the right of the interval P (b; a). Note that χ is a crisp binary relation in the usual sense. In [2, 3], conditions on R are explored such that the resulting crisp strict preference relation, χ is complete, transitive etc. In [4], it is shown that finite interval orders can be rationalized by a probabilistic choice function. Among many conditions one can impose on the fuzzy weak preference relation, the TS-transitivity as defined in [3] can be extended to a more general form. Definition 2 A valued relation, R,onA is called TStransitive for a given De Morgan triple ht; S; Ni if and only if for all a; b; c 2 A if T (R(a; b);n(r(b; a))) S(R(b; a);n(r(a; b))) and T (R(b; c);n(r(c; b))) S(R(c; b);n(r(b; c))) then T (R(a; c);n(r(c; a))) S(R(c; a);n(r(a; c))) Furthermore, if all the inequalities are strict, it is called strictly TS-transitive. Whenever T = min and S = max the property is called min-max transitivity. In [3], it is proved that, TS-transitivity is necessary and sufficient for the crisp strict preference, χ to be transitive: Proposition 1 Whenever ht; S; Ni is a continuous De Morgan triple, the crisp ordering relation χ as defined in (2) is transitive if and only if the valued relation R is T S-transitive (cf. Definition 2). Note that, well known transitivity conditions like mintransitivity and T -transitivity implies T S-transitivity and hence are sufficient for the transitivity of χ. The condition of TS-transitivity can be extended to TS-interval property as given in the following:

3 Definition 3 A fuzzy relation, R, on A is called to satisfy TS-interval property for a given continuous De Morgan triple ht; S; N i if and only if for all a; b; c; d 2 A if T (R(a; b);n(r(b; a))) S(R(b; a);n(r(a; b))) and T (R(c; d);n(r(d; c))) S(R(d; c);n(r(c; d))) then T (R(a; d);n(r(d; a))) S(R(d; a);n(r(a; d))) or T (R(c; b);n(r(b; c))) S(R(c; b);n(r(b; c))) Furthermore, if all the inequalities are strict, the property is called strict TS-interval property. TS-interval order property clearly generalizes TStransitivity property. Then, using Proposition 1, the following result is immediate: Corollary 1 A crisp binary relation, χ on A as defined in (2) is an interval ordering iff the underlying fuzzy relation R satisfies T S-interval property. Proof: We need to show that for all a; b; c; d 2 A: a χ b and c χ d =) a χ d or c χ b holds. Necessity and sufficiency immediately follows from Proposition 1. Λ Hence, S = (A; χ) is an interval order. Note that we have chosen to construct interval order using an irreflexive relation χ and hence the completeness condition is: 8a; b 2 A and a 6= b a χ b or b χ a: 3 Completing Interval Orders In this section we address the problem of aggregating preference orders of multiple agents where each agent can submit a partial preference order about a set of alternatives in A. Let S =(A; χ) be a partially ordered set. An element a in A is a minimal element if there is no b 2 A such that a χ b. The set of all minimal elements of S is denoted by Min(S). In general any partially ordered set (poset), S = (A; χ) can be completed (linearized) to L =(A; χ L ) using the following scheme: Step 0 L =? Step i for all a i 2 A do choose (a i 2 Min(S)) L ψ L + fa i g S ψ S nfa i g Output L The procedure choose is intentionally left vague here since with different instantiations of the procedure choose, one can come up with different extensions of a poset. In [13], Yager comes up with a similar greedy algorithm for aggregating the preferences of multiple agents reporting linear orders. We are going to use the same approach for agents reporting partial orders in Section 4. There has to be a performance measure for different aggregation methods. Definition 4 Let L = a 1 ;a 2 ; ;a n be a linear extension of S. Two consecutive elements a i ;a i+1 of L are separated by a jump if and only if a i and a i+1 are incomparable in S. The pair (a i ;a i+1 ) of L is called a bump. The total number of jumps of L is denoted by s(l). The jump number, s(s) of S is the minimum number of jumps in any linear extension, i.e., s(s) =minfs(l) :L is a linear extension of Sg The problem compute s(s) and find an optimal linear extension has been shown to be NP-hard (See [8] for references and heuristic algorithms). A particularly important extension algorithm is the greedy algorithm which requires the definition of Succ(a) on S =(A; χ) as: Succ(a) =fb 2 A : b χ ag: In an interval order, the sets of successors of any two elements a; b 2 A are related as either Succ(a) Succ(b) or Succ(a) Succ(b) [9]. So one can introduce: SMin(S) =fa 2 A : Succ(a) Succ(b) for all b 2 Ag: Then the extended greedy algorithm is given as: Greedy Extension Algorithm Step 0 L =?

4 Step i for all a i 2 A do if SMin(S) Succ(a i 1) 6=? choose(a i 2 SMin(S) Succ(a i 1)) else choose(a i 2 SMin(S)) L ψ L + fa i g S ψ S nfa i g Output L Of course, this is only a heuristic algorithm in terms of the jump number problem. f d e a b c c a e f b c e a d f b d a b Figure 1: Two interval orders and their (Hasse) diagrams 4 Aggregating importance weighed interval orders We assume that agents are (linearly) ordered with respect to a priority (as in [13]). This priority ordering may reflect a certain measure of confidence or reliability of agents. In aggregating interval orders, one can first extend all individual interval orders to their linear counterparts and then use a greedy algorithm as in [13] to fuse them. Another approach can be to fuse interval orders directly to obtain an aggregate linear order. This is the approach we will pursue in this paper. We hope that the aggregated order preserves as many relations from the individual orders as possible. The algorithm we propose below is in the spirit of the extended greedy algorithm of previous section generalized for multiple partial orders. Without loss of generality assume that we are aggregating m partial d e c f orders, S j, ordered in terms of their importance as fs 1 ;S 1 ; ;S m g. Greedy Fusion Algorithm Step 0 L =? X = A Step i j =1 L ψ L + fa i g X ψ X nfa i g Output L for all a i 2 A do if T j SMin X (S j ) 6=? then choose(a i 2 T j SMin X (S j ) else choose(a i 2 SMin X (S j )) j ψ j +1 if j = m +1then j ψ 1 where SMin X (S) is the same SMin function restricted to reference set X. The choose function in this algorithm is arbitrary. It can be a simple rule like FIFO or random selection. This, of course, is a greedy aggregation of interval orders. The resulting linear order is an extension of each and every individual interval order. The greedy fusion has desired characteristics of an aggregation function as shown in the following: Proposition 2 (Monotonicity) If an alternative a is preferred to b in all individual orders, a is also preferred to b in the aggregate order. Proof: If a χ i b in all individual orders then the greedy aggregation, by construction, cannot select a before b.λ Monotonicity is also known as Unanimity or the Pareto condition. A useful corollary is the idempotency. Corollary 2 (Idempotency) If all the individual preference orders are the same, the resulting aggregate preference order is also this same order. Proposition 3 (Positive Association) Assume S 1 ; ;S m are a collection of interval orders which result in an aggregate preference order S where a χ S b. Let S 0 1 ; ;S0 m be another set of interval orders where Si 0 differs from S i only in a being moved upward and/or b being moved downward in Si 0.IfS 0 is the aggregation of Si 0, then it must be the case that a χ S 0 b:

5 Proof: The nature of the algorithm does not allow such a to be selected (either as an intersection of the SMin function or from the SMin of an individual order) before b.λ Notice that there is no chance of a dictator in this setting since the aggregation is not based on voting but works over all agents in a given priority ordering. In this setting, no single ordering can alone be decisive between the preference on any a and b. For the two interval orders of Figure 1, the sets of successors are as given in Figure 2. Alternative Succ. in S 1 Succ. in S 2 a fb; c; e; fg fc; dg b fc; eg fc; dg c fg fg d fc; eg fg e fg fa; c; dg f fg fa; b; c; dg Figure 2: Successors of alternatives for the example Initially, SMin(S 1 )=fag and SMin(S 2 )=ffg. The resulting aggregate order is 1 : S = fc; d; e; b; f; ag: This order has one conflict and one bump with S 1 and one conflict and two bumps with S 2. Although the algorithm requires that agents are linearly ordered with respect to some sort of a priority, this information is only used whenever the intersection of SMin sets for all interval orders does not contain a common element. This algorithm is certainly in the spirit of conjunction or anding partial orders. A perfectly symmetric development is possible for an algorithm that results in the disjunction of posets. Since the algorithm is greedy, it does not consider the jump number problem of the individual orders. If the individual orders are first extended optimally and then aggregated using the greedy algorithm, the result could be better in terms of jump number problem of the aggregated order with each individual interval orders. For the example problem, optimal extensions of S 1 and S 2 are ejfjcbajd, and cjdaejbf, respectively 1 Note that the algorithm outputs L = fa; f; b; e; d; cg, a reverse order! where j denotes a jump. Greedy aggregation of these two linear orders is fc; e; b; a; f; dg, which results in two jumps and one conflict with S 1 and S 2. 5 Conclusion We have taken up the problem of fusing interval strict preferences. Interval strict preferences can arise in interval-valued preference structures utilizing the separation of the Boolean normal forms in fuzzy set theory. When the underlying fuzzy weak preference relation satisfies a condition called the TS-interval order property, the resulting crisp strict preference induces an interval order on the set of alternatives. This is essentially how fuzziness, in an ordinal sense, is captured in this framework. Although the interval-valued strict preference quickly gives way to a crisp binary relation in the classical sense, for the underlying fuzzy binary relation R a common condition like min-transitivity, or min-max interval property is sufficient for the resulting crisp order to also be transitive. When many interval orders (however constructed) are reported by many agents, the problem of fusing these orders to obtain a common ordering is also taken up. A greedy algorithm is suggested for aggregation. This algorithm takes each and every interval ordering report from the agents and successively constructs an aggregated linear order. This method satisfies certain properties of such aggregation functions. The preferences of many agents reporting different interval orders can be fused simply and efficiently in this setting without the undesirable effects of impossibility results. What remains to be investigated is, as the example in Section 4 shows, there is room for improvement in suggesting a modified aggregation algorithm in terms of the jump number problem. One particular approach can be to aggregate the underlying weak fuzzy relations of different agents first, come up with a aggregate fuzzy relation and then come up with a crisp ordering from that aggregate fuzzy relation. Conditions under which two approaches yield the same ordering is interesting and is left as future work. Acknowledgements This work is supported in part by BU Research Fund

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